where \(w_k \in \mathbb {R}^d\) , \(b_k, a_k \in \mathbb {R}\)  are the parameters for the \(k\) -th hidden neuron, \(\sigma \)  is the activation function.
Denote by \(W^{NN}_{r,d}\)  the class of functions with continuous partial derivatives of order \(\le r\)  with certain assumptions (see [1]} for detail).
Correspondingly, a hierarchical binary tree network is a network with the structure of
\(\begin{split}f(x_1, \ldots , x_d) = & h_{l1}(h_{(l-1), 1}(\ldots (h_{11}(x_1, x_2), \\&h_{12}(x_3, x_4), \ldots ), h_{(l-1), 2}(\ldots )),\end{split}\) 
